Special Relativity: Tensor Calculus and Four-Vectors
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
Topology and Physics 2019 - Lecture 2
Topology and Physics 2019 - lecture 2 Marcel Vonk February 12, 2019 2.1 Maxwell theory in differential form notation Maxwell's theory of electrodynamics is a great example of the usefulness of differential forms. A nice reference on this topic, though somewhat outdated when it comes to notation, is [1]. For notational simplicity, we will work in units where the speed of light, the vacuum permittivity and the vacuum permeability are all equal to 1: c = 0 = µ0 = 1. 2.1.1 The dual field strength In three dimensional space, Maxwell's electrodynamics describes the physics of the electric and magnetic fields E~ and B~ . These are three-dimensional vector fields, but the beauty of the theory becomes much more obvious if we (a) use a four-dimensional relativistic formulation, and (b) write it in terms of differential forms. For example, let us look at Maxwells two source-free, homogeneous equations: r · B = 0;@tB + r × E = 0: (2.1) That these equations have a relativistic flavor becomes clear if we write them out in com- ponents and organize the terms somewhat suggestively: x y z 0 + @xB + @yB + @zB = 0 x z y −@tB + 0 − @yE + @zE = 0 (2.2) y z x −@tB + @xE + 0 − @zE = 0 z y x −@tB − @xE + @yE + 0 = 0 Note that we also multiplied the last three equations by −1 to clarify the structure. All in all, we see that we have four equations (one for each space-time coordinate) which each contain terms in which the four coordinate derivatives act. Therefore, we may be tempted to write our set of equations in more \relativistic" notation as ^µν @µF = 0 (2.3) 1 with F^µν the coordinates of an antisymmetric two-tensor (i. -
1 Euclidean Vector Space and Euclidean Affi Ne Space
Profesora: Eugenia Rosado. E.T.S. Arquitectura. Euclidean Geometry1 1 Euclidean vector space and euclidean a¢ ne space 1.1 Scalar product. Euclidean vector space. Let V be a real vector space. De…nition. A scalar product is a map (denoted by a dot ) V V R ! (~u;~v) ~u ~v 7! satisfying the following axioms: 1. commutativity ~u ~v = ~v ~u 2. distributive ~u (~v + ~w) = ~u ~v + ~u ~w 3. ( ~u) ~v = (~u ~v) 4. ~u ~u 0, for every ~u V 2 5. ~u ~u = 0 if and only if ~u = 0 De…nition. Let V be a real vector space and let be a scalar product. The pair (V; ) is said to be an euclidean vector space. Example. The map de…ned as follows V V R ! (~u;~v) ~u ~v = x1x2 + y1y2 + z1z2 7! where ~u = (x1; y1; z1), ~v = (x2; y2; z2) is a scalar product as it satis…es the …ve properties of a scalar product. This scalar product is called standard (or canonical) scalar product. The pair (V; ) where is the standard scalar product is called the standard euclidean space. 1.1.1 Norm associated to a scalar product. Let (V; ) be a real euclidean vector space. De…nition. A norm associated to the scalar product is a map de…ned as follows V kk R ! ~u ~u = p~u ~u: 7! k k Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.2 1.1.2 Unitary and orthogonal vectors. Orthonormal basis. Let (V; ) be a real euclidean vector space. De…nition. -
Tensor Manipulation in GPL Maxima
Tensor Manipulation in GPL Maxima Viktor Toth http://www.vttoth.com/ February 1, 2008 Abstract GPL Maxima is an open-source computer algebra system based on DOE-MACSYMA. GPL Maxima included two tensor manipulation packages from DOE-MACSYMA, but these were in various states of disrepair. One of the two packages, CTENSOR, implemented component-based tensor manipulation; the other, ITENSOR, treated tensor symbols as opaque, manipulating them based on their index properties. The present paper describes the state in which these packages were found, the steps that were needed to make the packages fully functional again, and the new functionality that was implemented to make them more versatile. A third package, ATENSOR, was also implemented; fully compatible with the identically named package in the commercial version of MACSYMA, ATENSOR implements abstract tensor algebras. 1 Introduction GPL Maxima (GPL stands for the GNU Public License, the most widely used open source license construct) is the descendant of one of the world’s first comprehensive computer algebra systems (CAS), DOE-MACSYMA, developed by the United States Department of Energy in the 1960s and the 1970s. It is currently maintained by 18 volunteer developers, and can be obtained in source or object code form from http://maxima.sourceforge.net/. Like other computer algebra systems, Maxima has tensor manipulation capability. This capability was developed in the late 1970s. Documentation is scarce regarding these packages’ origins, but a select collection of e-mail messages by various authors survives, dating back to 1979-1982, when these packages were actively maintained at M.I.T. When this author first came across GPL Maxima, the tensor packages were effectively non-functional. -
SPINORS and SPACE–TIME ANISOTROPY
Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics. -
Physics 200 Problem Set 7 Solution Quick Overview: Although Relativity Can Be a Little Bewildering, This Problem Set Uses Just A
Physics 200 Problem Set 7 Solution Quick overview: Although relativity can be a little bewildering, this problem set uses just a few ideas over and over again, namely 1. Coordinates (x; t) in one frame are related to coordinates (x0; t0) in another frame by the Lorentz transformation formulas. 2. Similarly, space and time intervals (¢x; ¢t) in one frame are related to inter- vals (¢x0; ¢t0) in another frame by the same Lorentz transformation formu- las. Note that time dilation and length contraction are just special cases: it is time-dilation if ¢x = 0 and length contraction if ¢t = 0. 3. The spacetime interval (¢s)2 = (c¢t)2 ¡ (¢x)2 between two events is the same in every frame. 4. Energy and momentum are always conserved, and we can make e±cient use of this fact by writing them together in an energy-momentum vector P = (E=c; p) with the property P 2 = m2c2. In particular, if the mass is zero then P 2 = 0. 1. The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time di®erence between the events according to an observer moving at u = 0:8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0:8c. Answer: According to the formula for a Lorentz transformation, ³ u ´ 1 ¢tobserver = γ ¢tEarth-Sun ¡ ¢xEarth-Sun ; γ = p : c2 1 ¡ (u=c)2 Plugging in the numbers gives (notice that the c implicit in \light-minute" cancels the extra factor of c, which is why it's nice to measure distances in terms of the speed of light) 2 min ¡ 0:8(8:3 min) ¢tobserver = p = ¡7:7 min; 1 ¡ 0:82 which means that according to the observer, event B happened before event A! If we reverse the sign of u then 2 min + 0:8(8:3 min) ¢tobserver 2 = p = 14 min: 1 ¡ 0:82 2. -
General Relativity Fall 2019 Lecture 11: the Riemann Tensor
General Relativity Fall 2019 Lecture 11: The Riemann tensor Yacine Ali-Ha¨ımoud October 8th 2019 The Riemann tensor quantifies the curvature of spacetime, as we will see in this lecture and the next. RIEMANN TENSOR: BASIC PROPERTIES α γ Definition { Given any vector field V , r[αrβ]V is a tensor field. Let us compute its components in some coordinate system: σ σ λ σ σ λ r[µrν]V = @[µ(rν]V ) − Γ[µν]rλV + Γλ[µrν]V σ σ λ σ λ λ ρ = @[µ(@ν]V + Γν]λV ) + Γλ[µ @ν]V + Γν]ρV 1 = @ Γσ + Γσ Γρ V λ ≡ Rσ V λ; (1) [µ ν]λ ρ[µ ν]λ 2 λµν where all partial derivatives of V µ cancel out after antisymmetrization. σ Since the left-hand side is a tensor field and V is a vector field, we conclude that R λµν is a tensor field as well { this is the tensor division theorem, which I encourage you to think about on your own. You can also check that explicitly from the transformation law of Christoffel symbols. This is the Riemann tensor, which measures the non-commutation of second derivatives of vector fields { remember that second derivatives of scalar fields do commute, by assumption. It is completely determined by the metric, and is linear in its second derivatives. Expression in LICS { In a LICS the Christoffel symbols vanish but not their derivatives. Let us compute the latter: 1 1 @ Γσ = @ gσδ (@ g + @ g − @ g ) = ησδ (@ @ g + @ @ g − @ @ g ) ; (2) µ νλ 2 µ ν λδ λ νδ δ νλ 2 µ ν λδ µ λ νδ µ δ νλ since the first derivatives of the metric components (thus of its inverse as well) vanish in a LICS. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
Scalar-Tensor Theories of Gravity: Some Personal History
history-st-cuba-slides 8:31 May 27, 2008 1 SCALAR-TENSOR THEORIES OF GRAVITY: SOME PERSONAL HISTORY Cuba meeting in Mexico 2008 Remembering Johnny Wheeler (JAW), 1911-2008 and Bob Dicke, 1916-1997 We all recall Johnny as one of the most prominent and productive leaders of relativ- ity research in the United States starting from the late 1950’s. I recall him as the man in relativity when I entered Princeton as a graduate student in 1957. One of his most recent students on the staff there at that time was Charlie Misner, who taught a really new, interesting, and, to me, exciting type of relativity class based on then “modern mathematical techniques involving topology, bundle theory, differential forms, etc. Cer- tainly Misner had been influenced by Wheeler to pursue these then cutting-edge topics and begin the re-write of relativity texts that ultimately led to the massive Gravitation or simply “Misner, Thorne, and Wheeler. As I recall (from long! ago) Wheeler was the type to encourage and mentor young people to investigate new ideas, especially mathe- matical, to develop and probe new insights into the physical universe. Wheeler himself remained, I believe, more of a “generalist, relying on experts to provide details and rigorous arguments. Certainly some of the world’s leading mathematical work was being done only a brief hallway walk away from Palmer Physical Laboratory to Fine Hall. Perhaps many are unaware that Bob Dicke had been an undergraduate student of Wheeler’s a few years earlier. It was Wheeler himself who apparently got Bob thinking about the foundations of Einstein’s gravitational theory, in particular, the ever present mysteries of inertia. -
SEMISIMPLE WEAKLY SYMMETRIC PSEUDO–RIEMANNIAN MANIFOLDS 1. Introduction There Have Been a Number of Important Extensions of Th
SEMISIMPLE WEAKLY SYMMETRIC PSEUDO{RIEMANNIAN MANIFOLDS ZHIQI CHEN AND JOSEPH A. WOLF Abstract. We develop the classification of weakly symmetric pseudo{riemannian manifolds G=H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G=H and the signature of the invariant pseudo{riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature (n − 1; 1) and trans{lorentzian signature (n − 2; 2). 1. Introduction There have been a number of important extensions of the theory of riemannian symmetric spaces. Weakly symmetric spaces, introduced by A. Selberg [8], play key roles in number theory, riemannian geometry and harmonic analysis. See [11]. Pseudo{riemannian symmetric spaces, including semisimple symmetric spaces, play central but complementary roles in number theory, differential geometry and relativity, Lie group representation theory and harmonic analysis. Much of the activity there has been on the Lorentz cases, which are of particular interest in physics. Here we study the classification of weakly symmetric pseudo{ riemannian manifolds G=H where G is a semisimple Lie group and H is a reductive subgroup. We apply the result to obtain explicit listings for metrics of riemannian (Table 5.1), lorentzian (Table 5.2) and trans{ lorentzian (Table 5.3) signature. The pseudo{riemannian symmetric case was analyzed extensively in the classical paper of M. Berger [1]. Our analysis in the weakly symmetric setting starts with the classifications of Kr¨amer[6], Brion [2], Mikityuk [7] and Yakimova [16, 17] for the weakly symmetric riemannian manifolds. -
Integrating Electrical Machines and Antennas Via Scalar and Vector Magnetic Potentials; an Approach for Enhancing Undergraduate EM Education
Integrating Electrical Machines and Antennas via Scalar and Vector Magnetic Potentials; an Approach for Enhancing Undergraduate EM Education Seemein Shayesteh Maryam Rahmani Lauren Christopher Maher Rizkalla Department of Electrical and Department of Electrical and Department of Electrical and Department of Electrical and Computer Engineering Computer Engineering Computer Engineering Computer Engineering Indiana University Purdue Indiana University Purdue Indiana University Purdue Indiana University Purdue University Indianapolis University Indianapolis University Indianapolis University Indianapolis Indianapolis, IN Indianapolis, IN Indianapolis, IN Indianapolis, IN [email protected] [email protected] [email protected] [email protected] Abstract—This Innovative Practice Work In Progress paper undergraduate course. In one offering the course was attached presents an approach for enhancing undergraduate with a project using COMSOL software that assisted students Electromagnetic education. with better understanding of the differential EM equations within the course. Keywords— Electromagnetic, antennas, electrical machines, undergraduate, potentials, projects, ECE II. COURSE OUTLINE I. INTRODUCTION A. Typical EM Course within an ECE Curriculum Often times, the subject of using two potential functions in A typical EM undergraduate course requires Physics magnetism, the scalar magnetic potential function, ܸ, and the (electricity and magnetism) and Differential equations as pre- vector magnetic potential function, ܣ (with zero current requisite